Applied Analysis REU
Research Experiences for Undergraduates
June 1 - July 26, 2025
Research Experiences for Undergraduates
June 1 - July 26, 2025
All the carefully designed projects are in the area of applied mathematics, emphasizing the applications of the mathematical tools that an undergraduate student can master.
Projects are computational, theoretical, or a combination of these. The required techniques are mostly covered in calculus and differential equations undergraduate courses. Weeks one and two consist of intensive courses on dynamical systems, one-dimensional conservation laws, basic measure theory, and one-dimensional optimal transport.
Possible Topics Offered
Faculty Mentor: Charis Tsikkou
Conservation laws, which have important applications in gas dynamics, fluid mechanics, elasticity, aircraft aerodynamics, geophysical fluid dynamics, meteorology, general relativity, bio-sciences etc, state that the total amount of a certain quantity, such as mass, momentum, or energy, will not change in time during physical processes and can be written as a system of PDEs in divergence form. More generally, we consider systems of balance laws, which stipulate that the time rate of change in the amount of a quantity inside a domain is balanced by the rate of flux of this quantity through the domain’s boundary together with the rate of its production inside the domain.
The global-in-time existence of solutions to an initial value problem (Cauchy problem) is of particular interest. A fundamental feature of hyperbolic systems of balance laws is the development of discontinuities or shocks (infinite gradient) in finite time, even starting from smooth initial values. Because of this, solutions in the classical sense could fail to exist.
When studying multi-dimensional problems, it is common to consider simpler solutions with planar, cylindrical, and spherical symmetry, which appear in systems modeling the early stages of a gas flow between fragments of a bomb explosion with a spherical or cylindrical casing, nuclear explosions, etc. Seeking symmetric solutions has the advantage of changing a multi-d problem to a 1-dimensional one. Another common practice is seeking self-similar solutions, a common viewpoint in fluid mechanics. At this stage, we pursue a theory of the existence of solutions for a simpler class of initial data to models used in applications to develop methods that can provide insight into the stability and behavior of solutions. Such results can be used as building blocks in generalized schemes when solving Cauchy problems globally with large data.
The basic building block towards the proof of existence theorems of the Cauchy problem of conservation laws in one space dimension with small data is the solution to the Riemann problem, an initial value problem that consists of data containing two constant states, separated by a discontinuity at the origin which gives self-similar solutions. The solution is synthesized by constant states, shocks joining constant states that satisfy the Rankine-Hugoniot relation and/or centered rarefaction waves bordered by constant states or contact discontinuities. The main focus of the projects on this topic is the study of models whose solutions to the Riemann problems involve the so-called singular shocks, which are a more compressive generalization of the ordinary shock wave where at least one state variable develops an extreme concentration in the form of a weighted Dirac delta function. Constructing Riemann solutions is essential from both analytical and numerical points of view.
More specifically, we considered a system of two balance laws of Keyfitz-Kranzer type with varying generalized Chaplygin gas, which exhibits negative pressure and is a product of a function of time and the inverse of a power of the density. The Chaplygin gas is a fluid designed to accommodate measurements for the early universe and late-time universal expansion while obeying the pressure-density-time relation. To the best of our knowledge, there has been little research on combining both the varying and generalized model, denoted by the varying generalized Chaplygin gas model (VGCG).
We produced an explanation and description of the non-self-similar Riemann solutions, including the non-classical singular solutions. We also found that due to a direct dependence on time, a change in the regions allows for combinations of classical and non-classical singular solutions to occur. Therefore, a Riemann solution can have different solutions over several time intervals. Our findings were confirmed numerically using the local Lax-Friederichs scheme. To the best of our knowledge, direct time dependence resulting in changes in the areas where classical and non-classical singular solutions exist has not been analyzed and confirmed numerically before.
The question now arises: Can singular shock solutions be predicted within a system? Can we find a physical interpretation of their significance, explain how they satisfy the equations, and establish a better definition encompassing a broader range of examples? Is there a connection between singular shocks, genuinely nonlinear systems, and change of type (conservation laws that are not everywhere hyperbolic)? To this end, we propose the following problems:
Faculty Mentor: Adrian Tudorascu
The attractive pressureless Euler-Poisson system in spatial dimension one describes the evolution of a mass distribution moving initially at a prescribed velocity and subject to gravitational interaction. It consists of the conservation of mass equation (continuity equation) and the conservation of momentum equation:
\begin{equation}\label{PEP}\tag{PEP}
\begin{cases}
\partial_t \rho + \partial_y (\rho v) = 0 \\
\partial_t (\rho v) + \partial_y (\rho v^2) = -\frac{1}{2}(\mathrm{sgn}\ast\rho)\rho.
\end{cases}
\end{equation}
Here $\mathrm{sgn}$ is the signum function, defined as
\[ \mathrm{sgn}(y) = \begin{cases}
-1 &\mbox{ if } y < 0, \\
\\ 0 &\mbox{ if } y=0, \\
\\ 1 &\mbox{ if }y>0.
\end{cases}
\]
This system governs the dynamics of a collection of particles in which the force on each particle is proportional to the total mass to the right of the particle minus the total mass to the left of the particle. We are looking for a special type of solutions, called Sticky Particles solutions (SPS); what this means is that when particles collide, they undergo perfectly inelastic collisions so they stick together to form a new particle whose mass equals the sum of all the particles involved in the collision. Conservation of momentum through collisions is also imposed, i.e. the newly formed particle starts off at the time of collision with a momentum equal to the sum of the momenta of all the participating particles right before the collision. This model was first introduced by Zeldovich in 1970 as a cosmological model to study the formation of large structures in the universe. When there is no gravitational term on the right-hand side of the momentum equation, the corresponding system is called the pressureless Euler system. In that case, the model can be more precisely described as follows: if $m_i$, $i=1,...,n$ is a discrete system of masses initially located at $-\infty < y_1 < ... < y_n < +\infty$ and moving with initial velocities $v_i$, $i=1,...,n$, then one assumes that the velocities remain constant while there is no collision. At the collision of a group of particles, the particles stick together, and the conservation of momentum gives the initial velocity of the newly formed particle. It turns out that the pressureless Euler system describes the evolution of this system. In most of the works on the existence of solutions, the initial distribution is approximated by averages of Dirac masses, and the ensuing Sticky Particles system is used to approximate solutions to the pressureless Euler system (PE).
During the last 25 years or so, PE has been studied by a host of authors and by different techniques that include the Sticky Particles model, a description of the problem by an alternative scalar conservation law problem, and a semigroup approach. The class of solutions obtained by approximating the initial mass distribution by convex combinations of Dirac masses is stable and, naturally, contains the discrete Sticky Particles solutions; thus, we call such solutions SPS (Sticky Particles Solutions).
Recently, A. Tudorascu employed SPS and a reflection principle to solve the problem in the case where the evolution is confined to an arbitrary closed subset $\mathcal{C}$ of the real line by imposing sticky boundary conditions. Soon after R. Hynd and A. Tudorascu showed that if $\mathcal{C}$ is compact, then the SPS converges asymptotically to an equilibrium solution. Whereas a uniform rate of decay of $1/t$ for the velocity is easily obtained under the assumption that the SPS remains confined to a compact set for all time, in their paper they left open the problem of the rate of decay for the mass distribution. They did conjecture, encouraged in most part by numerical simulations, that a rate of convergence did not exist; in other words, the mass distribution can (depending on the initial conditions) converge to equilibrium arbitrarily slowly. This problem was the research focus in [3] during the summer of 2024. The focus in [2] was the attractive pressureless Euler-Poisson system \eqref{PEP}. While it was expected that the gravitational pull between particles would speed up the rate of decay to equilibrium, there were many technical challenges to overcome. The analysis performed by R. Hynd and A. Tudorascu for the asymptotic behavior of SPS to the PE system was heavily reliant on the Lagrangian description of the SPS in terms of projections of a map involving the initial mass distribution and velocity onto the convex cone of nondecreasing square integrable random variables. The asymptotic behavior of SP solutions to \eqref{PEP} is very different and that should come as no surprise given that the attractive gravitational interaction between particles has the net effect of bringing all the masses together long term. No confinement to compact subsets is now necessary, as it has been proven that for any initial distribution $\rho_0$ with finite second moment and any $v_0\in L^\infty(\rho_0)$, the SPS corresponding to this initial data approaches asymptotically in the 1-Wasserstein distance the rectilinear trajectory of particles of mass one originating at the center of mass of $\rho_0$ and moving at a constant velocity equal to the average of $v_0$ with respect to $\rho_0$. The rate of decay is no slower than $1/t$. It was also shown that the SP solution collapses into the equilibrium in finite time if and only if the support of $\rho_0$ is bounded.
If the negative sign is removed from the momentum equation in \eqref{PEP}, we end up with the ${\it repulsive}$ pressureless Euler-Poisson system. It is natural that, in this case, the particles tend to repulse each other and not stick together. The sticky condition can still be imposed, and conditions on the initial velocities will now be more restrictive in order for collisions to occur. We propose the following problems:
1. Given an initial distribution of particles supported in a compact interval, identify conditions (necessary, sufficient?) on the initial velocities under which the SPS stays in a given compact set for all time. Is it true that this can only happen if all particles coalesce into a stationary one in finite time? Are there configurations in which the coalescence is asymptotic?
2. It stands to reason that for all configurations discussed above the solution converges asymptotically to an equilibrium solution. Is it true that, just as in the attractive case, the equilibrium can only be a stationary Dirac mass?
3. What is the decay rate to equilibrium if the equilibrium exists?
Note, additional projects may be added to this list. If you have a special request, please contact the directors for help.
[1] J. Frew, N. Keyser, E. Kim, G. Paddock, C. Toumbleston, S. Wilson, and C. Tsikkou, “An analysis of a 2 × 2 Keyfitz–Kranzer type balance system with varying generalized Chaplygin gas,” Physics of Fluids, 36 (2024), 096132. https://doi.org/10.1063/5.0231413
[2] B. Becerra, J. Linderoth, H. Pesin, A. Tudorascu, and R. Wassink, “Sticky Particles solutions for the attractive pressureless Euler-Poisson system; a projection formula and asymptotic behavior,” submitted, 2024.
[3] A. Tudorascu, and R. Wassink, “Compactly supported Sticky Particles solutions decay to equilibrium arbitrarily slowly”, submitted, 2024.